Precalculus Examples

$x^{2} - {(3 - y)}^{2} = 24$

Step 1

Simplify the left side $x^{2} - {(3 - y)}^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(3 - y)}^{2}$ as $(3 - y) (3 - y)$ .

$x^{2} - ((3 - y) (3 - y)) = 24$

Step 1.1.2

Expand $(3 - y) (3 - y)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$x^{2} - (3 (3 - y) - y (3 - y)) = 24$

Step 1.1.2.2

Apply the distributive property.

$x^{2} - (3 \cdot 3 + 3 (- y) - y (3 - y)) = 24$

Step 1.1.2.3

Apply the distributive property.

$x^{2} - (3 \cdot 3 + 3 (- y) - y \cdot 3 - y (- y)) = 24$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $3$ by $3$ .

$x^{2} - (9 + 3 (- y) - y \cdot 3 - y (- y)) = 24$

Step 1.1.3.1.2

Multiply $- 1$ by $3$ .

$x^{2} - (9 - 3 y - y \cdot 3 - y (- y)) = 24$

Step 1.1.3.1.3

Multiply $3$ by $- 1$ .

$x^{2} - (9 - 3 y - 3 y - y (- y)) = 24$

Step 1.1.3.1.4

Rewrite using the commutative property of multiplication.

$x^{2} - (9 - 3 y - 3 y - 1 \cdot (- 1 y \cdot y)) = 24$

Step 1.1.3.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 1.1.3.1.5.1

Move $y$ .

$x^{2} - (9 - 3 y - 3 y - 1 \cdot (- 1 (y \cdot y))) = 24$

Step 1.1.3.1.5.2

Multiply $y$ by $y$ .

$x^{2} - (9 - 3 y - 3 y - 1 \cdot (- 1 y^{2})) = 24$

Step 1.1.3.1.6

Multiply $- 1$ by $- 1$ .

$x^{2} - (9 - 3 y - 3 y + 1 y^{2}) = 24$

Step 1.1.3.1.7

Multiply $y^{2}$ by $1$ .

$x^{2} - (9 - 3 y - 3 y + y^{2}) = 24$

Step 1.1.3.2

Subtract $3 y$ from $- 3 y$ .

$x^{2} - (9 - 6 y + y^{2}) = 24$

Step 1.1.4

Apply the distributive property.

$x^{2} - 1 \cdot 9 - (- 6 y) - y^{2} = 24$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Multiply $- 1$ by $9$ .

$x^{2} - 9 - (- 6 y) - y^{2} = 24$

Step 1.1.5.2

Multiply $- 6$ by $- 1$ .

$x^{2} - 9 + 6 y - y^{2} = 24$

Step 1.2

Reorder.

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Step 1.2.1

Move $- 9$ .

$x^{2} + 6 y - y^{2} - 9 = 24$

Step 1.2.2

Move $6 y$ .

$x^{2} - y^{2} + 6 y - 9 = 24$

Step 2

Set the equation equal to zero.

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Step 2.1

Subtract $24$ from both sides of the equation.

$x^{2} - y^{2} + 6 y - 9 - 24 = 0$

Step 2.2

Subtract $24$ from $- 9$ .

$x^{2} - y^{2} + 6 y - 33 = 0$

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